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mathmari
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Hey! ![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
We have that $D_n=\langle a,s\mid s^n=1=a^2, asa^{-1}=s^{-1}\rangle$.
I want to show the following:
1. $D_n'=\langle [x,y]\mid x,y\in D_n\rangle=\langle xyx^{-1}y^{-1}\mid x,y\in D_n\rangle$
We have that $a,s\in D_n$, so $[s,a]\in D_n' \Rightarrow sas^{-1}a^{-1}\in D_n' \Rightarrow sas^{-1}a\in D_n' \Rightarrow sas^{-1}a\in D_n' \Rightarrow saasa^{-1}a\in D_n' \Rightarrow s^2a^2\in D_n' \Rightarrow s^2\in D_n' $.
Could you give me some hints for 2., 3., 4. ? (Wondering)
We have that $D_n=\langle a,s\mid s^n=1=a^2, asa^{-1}=s^{-1}\rangle$.
I want to show the following:
- $s^2\in D_n'$
- $D_n'\cong \mathbb{Z}_n$ if $n$ is odd
- $D_n'\cong \mathbb{Z}_{\frac{n}{2}}$ if $n$ is even
- $D_n$ is nilpotent if and only if $n=2^k$ for some $k=1,2,\dots $
1. $D_n'=\langle [x,y]\mid x,y\in D_n\rangle=\langle xyx^{-1}y^{-1}\mid x,y\in D_n\rangle$
We have that $a,s\in D_n$, so $[s,a]\in D_n' \Rightarrow sas^{-1}a^{-1}\in D_n' \Rightarrow sas^{-1}a\in D_n' \Rightarrow sas^{-1}a\in D_n' \Rightarrow saasa^{-1}a\in D_n' \Rightarrow s^2a^2\in D_n' \Rightarrow s^2\in D_n' $.
Could you give me some hints for 2., 3., 4. ? (Wondering)